Download Lesson Title: Properties of Parallelogram

Name: ______________________________________ Date: __________ Period:_____
Investigating Parallelograms
Lesson Goals: To engage in discovering properties of parallelograms by plotting points on a
coordinate plane, calculating slopes, measuring angles and calculating distances of side
lengths. To conjecture about the relationship between diagonals in a parallelogram and how
they bisect one another.
Directions: Look around the room and try to find at least 5 examples of parallelograms.
Write and draw these items in the first and second columns of the table below, titled
INTRODUCTION. Please do not write in the third and fourth columns, titled CLOSURE.
INTRODUCTION
Item
1
2
3
4
5
Object name
CLOSURE
Picture
Is it a
parallelogram?
(yes / no)
Why or
why not?
Name: ______________________________________ Date: __________ Period:_____
Activity One
1. On a coordinate grid, plot and label the points A(6,2), B(17,5), C(13,12), and D(2,9).
2. What general shape will be formed if you connect A to B, B to C, C to D, and D to A?
Connect each pair of points to verify your answer.
3. Use the distance formula, d  ( x1  x2 )2  ( y1  y2 )2 , to determine the length of each
of the segments.
Length of AB
=
Length of BC
=
Length of CD
=
Length of DA
=
4. What can you conclude about the lengths of the segments in this figure?
Name: ______________________________________ Date: __________ Period:_____
Activity Two
1. On a coordinate grid, plot and label the points A(-7,5), B(-2,-4), C(8,-8), and D(3,1).
2. What general shape will be formed if you connect A to B, B to C, C to D, and D to A?
Connect each pair of points to verify your answer.
3. Use the Slope formula, m 
( y1  y2 )
, to determine the slopes of each of the segments.
( x1  x2 )
Slope of AB
=
Slope of BC
=
Slope of CD
=
Slope of DA
=
4. What can you conclude about the slopes of the segments in this figure?
Name: ______________________________________ Date: __________ Period:_____
Activity Three
1. On a coordinate grid, plot and label the points A(-3,0), B(-1,5), C(5,4), and D(3,-1).
2. What general shape will be formed if you connect A to B, B to C, C to D, and D to A?
Connect each pair of points to verify your answer.
3. Remember that diagonals are segments that connect two non-adjacent vertices of a
polygon. How many diagonals will the figure above have? Use a straightedge to draw
the diagonals and verify your answer.
Name: ______________________________________ Date: __________ Period:_____
 x  x   y  y 
4. Using the Midpoint formula, midpoint   1 2 , 1 2  , find the midpoint of
2
2


diagonal AC . Plot and label this midpoint on the above coordinate plane as point F.
5. Use the distance formula, d  ( x1  x2 )2  ( y1  y2 )2 , to determine the length of each
of these segments.
Length of AF
=
Length of CF
=
From these lengths, what can you verify?
6. Use the distance formula, d  ( x1  x2 )2  ( y1  y2 )2 , to determine the length of each
of these segments.
Length of BF
=
Length of DF
=
From these lengths, what can you conclude?
7. What can you conclude about the diagonals in this figure?
Name: ______________________________________ Date: __________ Period:_____
Activity Four
 Consecutive angles of a quadrilateral are two angles that share a common ray.
 Opposite angles of a quadrilateral are two angles that are located on either end of a
diagonal, they are directly across from one another.
Consider your Geoboard to be the first quadrant of the coordinate plane, as the picture
illustrates below.
y-axis
x-axis
Steps:
1. Assume that the bottom outside edge of the Geoboard is the x-axis and the left outside
edge is the y-axis. Stretch the rubber band around these 4 coordinates to create a
parallelogram.
A(1,1)
B(3,1)
C(3,5)
D(5,5)
2. Use a protractor to measure the angles at each of the vertices and record them below.
m A =
m B =
m C =
m D =
3. Do you notice any similarities or differences among the angle measurements? Explain.
Name: ______________________________________ Date: __________ Period:_____
4. Now stretch the rubber band around these 4 coordinates to create a different
parallelogram.
E(2,1)
F(2,4)
G(4,2)
H(4,5)
5. Use a protractor to measure the angles at each of the vertices and record them below
m E =
m F =
m G =
m H =
6. Do you notice any similarities or differences among the angle measurements in the
second parallelogram? Explain.
7. Based on your responses to the #1-6, what general statement(s) can you make about a
property or properties found in all parallelograms regarding angle measurements?
8. Using your protractor draw a parallelogram below using only the angle measurements
to guide you. Label the angle measurements in your parallelogram.
Name: ______________________________________ Date: __________ Period:_____
QUIZ
1. On a coordinate grid, plot and label the points A(-3,3), B(8,5), C(2,-1).
2. Using the information that you discovered in the previous activities, where could you place
point D in order to make a parallelogram?
3. Please verify your solution by showing the slopes of the sides, the lengths of the sides, the
midpoint of the diagonals as well as the angle measures.
4. After you have found one point that makes a parallelogram, is it possible to find another
point that will make a different parallelogram with the three given points? How many different
points exist that will form a parallelogram with the three given points? List as many of these
points as you can.